Webb has two aims, a polemical one and a historical one. His polemical aim is to vindicate mechanism against those who, like myself, have argued that Gödel's undecidability result makes mechanism untenable. His historical aim is to relate modern concepts of computability, effectiveness and recursiveness, and the controversies about them, with earlier controversies about formalism and finitism. Whatever verdict is given on the success of his polemical endeavour, his book is a valuable contribution to the history of mathematical philosophy. Webb has read widely and writes clearly. What will be of particular value to British readers, for whom the North Sea is still much wider than the Atlantic Ocean, is Webb's full and careful account of the development of the philosophy of mathematics in the minds of Gauss, Dedekind, Kronecker, Frege, Hilbert, Husserl and Bernays. Much as Euclid earlier had been compelled to lay down the parallel postulate as a separate axiom, so the failure to give a clear and coherent grounding of the concept of infinity in the laws of thought forced mathematicians in the -end to lay down axioms of infinity and of the continuum in order to be able to go on doing mathematics with a clear conscience.

That formalisation is the outcome of mathematicians' efforts to givea coherent account of their arguments is undoubtedly true, and that formalism is strongly imbued with finitist ideals is true also. That the history of mathematics gives support to formalism-and finitism is not so evidently true. Any critical enquiry that seeks to elucidate concepts and establish propositions to the satisfaction of a stringent sceptic is going to invoke a decision-procedure in which concepts can be effectively defined and propositions effectively proved in a finite number of clearly rule-governed steps. If mathematics is defined by the axiomatic method, then Webb's argument goes through, and he shows how the claim that all arguments should be expressed more geometrico was gradually refined into Church's [442] Thesis, which identifies effectiveness with recursiveness. But it is not to be taken for granted that mathematics is to be defined in terms of the axiomatic method. Many-notably to the readers of this Journal the late lmre Lakatos-have maintained that it is not. Webb does not discuss, nor in a fairly full bibliography even mention Lakatos' arguments, and although he refers to the difficulty of formalism that it has to regard axioms as simply postulates and not as being true, he does not give much space to the antiformalist tradition in modern mathematics. This is not a severe criticism, inasmuch as his objective is not to argue for formalism as such but to trace the development of the ideas of modern mechanism, because only by so doing could he understand the points at issue today. Many philosophers will applaud his apologia for his historical approach (p. xiii):

we cannot fully understand our own conceptual scheme without plumbing its historical roots, but in order to appreciate those roots we may well have to filter them back through our own ideas.

Webb's polemical aim is to vindicate mechanism. The immediate question is what he means by mechanism. He maintains that a machine need not be predictable in principle: our understanding of the physical universe has been expanded by thermodynamics, quantum mechanics and Turing's theorem so that we now have an `indeterministic mechanism' (p. 6, cf. p. 200). It seems a misnomer. Machines that are not even in principle predictable are not usefully described as machines at all, although they may be indubitably material. What Webb seems to be arguing is a soft materialism, opposed to the mentalism that Descartes and reputedly Gödel espoused, but not carrying any determinist overtones or posing any threat to the freedom of the will. Human behaviour can be described ex post facto in finite terms, but it cannot, and could not have been, predicted on the basis of finite descriptions available at the time. The past, what we actually have done, is fixed and definite: the future is open, as is also the range of potential might-have-beens, which we could have done, but did not. It is an interesting position, and quite possibly true but it does not stand in the tradition of Hobbes and La Mettric, and was never put in jeopardy by Gödel's incompleteness theorem.

Webb's real concern is not with the nature of man, but the nature of reason, and mechanism is for him the claim that every effective procedure can be simulated by a Turing machine. Of course, there are, notoriously some things a Turing machine cannot do-that is, some questions-e.g. the halting problem-for which there is no effective decision procedure. But mechanism is, on Webb's view, saved from disaster by Gödel's theorem which is `a kind of invisible protective shield encasing the foundation for modern mechanism' (p. g). Although diagonalization seems to offer a way whereby we can always transcend the limits of any antecedently specified set of functions or procedures, diagonalization itself can be effectively specified, and so included within the compass of effective functions or procedures. But [443] once we follow Kleene and admit partial functions (i.e. functions which are not defined for every number), then diagonalization will not always lead to a new effective function outside the compass of those already specified. And so, Webb argues, in spite of Gödel's theorem, or rather because of Gödel's theorem, we can maintain that all reasoning can be simulated by a Turing machine, even the recognition that some questions cannot be decided by an effective decision procedure.

Several doubts arise. Gödel's theorem is indeed very close to Turing's theorem, but it is hard to see it as a protective shield for Turing's thesis. Partial functions lack an air of effectiveness. Exactly what is meant by effectiveness is unclear, or why it should be the hallmark of cogent reasoning. The original point in the Gödelian argument against mechanism was not simply that there were some things which a particular Turing machine could not do-which a Turing machine is indeed capable of acknowledging-but there were some things which a Turing machine could not do and a mind could. Of these points the one about effectiveness is perhaps of the greatest philosophical interest. An effective procedure is a sure-fire one. If I have an effective procedure, I can get the right answer, or force you to admit that it is the right answer, after only a finite number of steps. Plato in the Phaedrus (263a) draws the distinction between descriptive arguments, which can be so settled, and evaluative ones which can drag on interminably. But although they can, they do not have to. I may find a moral argument cogent, although if I dig my toes in I cannot be compelled to acknowledge its cogency. Equally in mathematics, I may be able to hit on a convincing demonstration although I have no method of finding one, nor any antecedently specified prescription of what should count as a proof. We have, as we know, no effective method of listing all the non-theorems of the first-order predicate calculus. But if I were able to take say, von Neumann with me on my tour of the `British Museum' (p.189), he might well be able to produce for every non-theorem a demonstration, often of a different type in different cases, that it was indeed a non-theorem. Clearly, these demonstrations would not all be formal proofs within first-order predicate calculus, or they could be enumerated systematically. But they might be convincing none the less, just as Gödel's argument convinces us that the Gödelian sentence is true although not provable in the system. In that case we should have reasons, although no effective method, for ascribing nontheorem-hood. And, it seems to me, not only occasionally in mathematics but often in other disciplines we have no effective method for getting the right answers, but can in individual cases produce reasoned answers, different in each case, which commend themselves to our judgement.

Webb will have none of this. To have reasons is to have an effective method.

But these formulations make it clear that the real source of Lucas' feeling of superiority here is just the effectiveness with which he can find the Achilles' heel of any machine. This alone is what makes him confident (p. 230).

[444]

But one can feel confident without having an effective method within the meaning given to effective---i.e. programmable into a Turingmachine. Effectiveness is being used ina very strong sense when it is being explicated in terms of computability. But then it is not necessary that all reasoning must in this sense be effective. And in the sense in which reasoning might be necessarily effective, effectiveness does not imply computability.

In spite of these criticisms, Webb has advanced the argument. He may not carry conviction in his advocacy of Church's thesis and Turing's thesis, but has helped to make clear what isat stake, and what the considerations are on either side.